Renewal theory for uniform random variables

This project will focus on finding formulas for E[N(t)] using one of the classical problems in the discipline first, and then extending the scope of the problem to include overall times greater than the time t in the original problem. The expected values in these cases will be found using the uniform and exponential distributions of random variables

An analytical study of liquid outflow from cylindrical tanks during weightlessness Final report

Liquid outflow from cylindrical tanks during weightlessnes

Investigation of steady incompressible flows in the vicinity of an airfoil

O-type body-fitted coordinate systems are numerically generated about NACA0012 and NACA4412 airfoils. These coordinate systems are used to effect a coordinate transformation from the physical plane to a rectangular transformed plane. The software developed also has a coordinate contraction capability. Plots of uncontracted and contracted coordinate systems are presented. The nonconservative, velocity-pressure formulation of the two-dimensional, steady-state Euler equations are transformed to the rectangular plane. The boundary conditions at the airfoil are determined by the source-panel method, following which, the explicit finite difference form of the transformed Euler equations are solved using a space-marching scheme. Non-lifting potential flow solutions are obtained for the aforementioned airfoils at 0 and ±16 angles of attack. Pressure and velocity distribution plots are presented for all cases

A theoretical investigation of the aerodynamics of slender wing-body combinations exhibiting leading-edge separation

Theoretical investigation of aerodynamics of slender wing-body combinations exhibiting leading edge separatio

Modelling the cooking process of a single cereal grain

Four models are developed to assist with the uniform and accurate cooking of whole grains for Uncle Tobys breakfast cereals. 1. Heat satisfies a linear conduction equation and is found to rapidly penetrate the grain. 2. Moisture satisfies a non-linear diffusion equation, and is found to penetrate the grain more slowly than heat. The more sophisticated moisture diffusion model is solved by numerical and analytic techniques for spherical and ellipsoidal grains. A vital role is played in the moisture diffusion model by the concept of the mean action time for wetting a grain. These first two models are used to determine sensitivity to key cooking parameters, and to calculate the degree of over-cook in the existing batch steam process. Recommendations are made for improving and speeding up the cooking process. The last two models are modifications of the nonlinear moisture penetration model 2. above. The results of these improved models have the potential to provide finer adjustments to estimates of wetting times. 3. A cereal grain swells significantly during wetting. A model that takes this into account is developed and solved approximately. 4. Another wetting model describes the effect of the gelatinisation reaction, slowing moisture penetration, and leading to a sharp front entering the grain. The effect of gelatinisation on the speed of moisture penetration is expected to be more important for the present high-temperature cooking process, than when soaking a grain at a lower temperature. This model is also developed and solved approximately

Collective electronic effects in scanning probe microscopy

The surface plasmon dispersion relations are calculated for a metal coated dielectric probe above a dielectric half space with and without metal coating. Employing prolate spheroidal coordinate system this configuration was modeled as confocal single-sheeted hyperboloids of revolution superimposed on planar domains. The involved media are characterized by frequency dependent, spatially local dielectric functions. Due to subwavelength dimensions of the region of interest, nonretarded electrodynamics is utilized to derive exact analytical expressions describing the resonant surface modes. The dis-persion relations are studied as functions of the parameter that defines the hyperboloidal boundaries of the tip and the corresponding coating, and as functions of the involved coating thicknesses. Both parallel and perpendicular polarizations are considered. The results are simulated numerically and limiting cases are discussed with comparison to the Cartesian thin foil case. Using this new type of probe-substrate configuration, the surface plasmon coupling mechanism is investigated experimentally utilizing a scanning probe microscope, and the signal strength acquired by the probe is measured as a function of the distance between the probe and the sample. This is repeated at three different wavelengths of the incident p-polarized photons used to stimulate surface plasmons in the thin metal foil. The results are compared with the theory. Utilizing the prolate spheroidal coordinate system, the related and relevant problem of the Coulomb interaction of a dielectric probe tip with a uniform field existing above a semiinfinite, homogeneous dielectric substrate was studied. This is of interest in atomic force microscopy when the sample surface is electrically charged. The induced polarization surface charge density and the field distribution at the bounding surface of the dielectric medium with the geometry of a single-sheeted hyperboloid of revolution located above the dielectric half space interfaced with a uniform surface charge density is calculated. The force density on the hyperboloidal probe medium is calculated as a function of the probe tip shape. The potential and the field distributions are calculated in the neighborhood of the apex of the tip. The calculation is based on solving Laplace\u27s equation and employing a newly derived integral expansion for the vanishing dielectric limit of the potential. The integral expansion is analytically proved and numerically verified. The involved numerical simulations comprise the evaluation of infinite double integrals involving conical functions similar to those arising in the Mehler-Fock integral transform

Numerical simulation of separated flows

A new numerical method, based on the Vortex Method, for the simulation of two-dimensional separated flows, was developed and tested on a wide range of gases. The fluid is incompressible and the Reynolds number is high. A rigorous analytical basis for the representation of the Navier-Stokes equation in terms of the vorticity is used. An equation for the control of circulation around each body is included. An inviscid outer flow (computed by the Vortex Method) was coupled with a viscous boundary layer flow (computed by an Eulerian method). This version of the Vortex Method treats bodies of arbitrary shape, and accurately computes the pressure and shear stress at the solid boundary. These two quantities reflect the structure of the boundary layer. Several versions of the method are presented and applied to various problems, most of which have massive separation. Comparison of its results with other results, generally experimental, demonstrates the reliability and the general accuracy of the new method, with little dependence on empirical parameters. Many of the complex features of the flow past a circular cylinder, over a wide range of Reynolds numbers, are correctly reproduced